Problèmes inverse en optique non-imageante et équations au Jacobien généré.

Mots clé :

• optimal transport
• generated Jacobian equations.

This thesis is motivated by non imaging optics problems. To begin with, we introduce the
field of non-imaging optics. We see that some of the problems arising in this field can be
recast as optimal transport problems or more generally as generated Jacobian equations
(GJE). This work is divided in two parts.
The first part deals with the stability of solutions to optimal transport problems
under variation of the measures, and is closely related to the convergence of numerical
approaches to solve optimal transport problems and justifies many of the applications of
optimal transport. 
The second part deals with Generated Jacobian equations, that have been introduced
by Trudinger [Disc. cont. dyn. sys (2014), pp. 1663–1681] as a generalization of Monge-
Ampère equations arising in optimal transport. We present and study a damped Newton
algorithm for solving these equations in the semi-discrete setting, meaning that one of
the two measures involved in the problem is finitely supported and the other one is
absolutely continuous.

Directeurs:

• Boris Thibert
• Quentin Mérigot